George Osborne is preparing to set out his plans to return bailed-out Lloyds Banking Group and Royal Bank of Scotland to the private sector after the International Monetary Fund called on him to devise a "clear strategy" for the two banks.

The Washington-based body, in London to present its annual health check on the UK economy, also told the chancellor that if the two banks needed more capital to bolster their financial strength he should pour in more taxpayer funds, as it would prove beneficial to the economy.

Some £65bn of taxpayer money is already locked up in shares in RBS and Lloyds, which both issued stock market announcements to insist they did not need to tap investors – particularly taxpayers – for fresh funds to plug capital shortfalls, estimated to be about £10bn.

The intervention of the IMF forced Osborne to give the clearest indication yet he will outline his strategy for the two banks next month, with speculation focusing on his Mansion house speech in June.

More generally, the IMF said banks should be required to raise equity, cut dividends and show restraint on remuneration rather than cut back on lending.

Osborne said he would reveal his decision on Lloyds and RBS after the crucial report by the Parliamentary Commission on Banking Standards, which is expected to report next month . The report may call for full nationalisation of RBS, already 81% state owned. Lloyds is 39% owned by the taxpayer.

"Having refocused their business, now is the time for a clear strategy on how to return RBS and Lloyds to the private sector in a way that protects value for the taxpayer," Osborne said.

Shares in the two banks rose after they said their long-running discussions with the new City regulator, the Prudential Regulation Authority (PRA), over capital requirements ended. The banks said they could sell off businesses and cut down on risks rather than raise fresh funds to fill the shortfall.

Lloyds ended nearly 2p higher at 62.96p – above the 61p level the government now sees as break-even – and RBS ended 7.4p up at 349.6p.

The IMF presented a dilemma for Osborne by making clear that value to taxpayers should be central in any sell-off. Shares in both banks are firmly below levels the City regards as break-even: 73.6p for Lloyds and 502p for RBS, levels leaving taxpayers with £17bn of losses..

"Any strategy should seek to return the banks to private hands in a way that maximises the value for taxpayers, strengthens confidence and competition in the sector, and minimises outward spillovers," the IMF said as it indicated a strategy should be outlined by the end of the year. "In this context, if a sovereign backstop is required to meet a capital shortfall, it should be provided, as this would have a high multiplier."

The IMF did not indicate when stakes should be sold off and noted that "challenges remain" as the banks had failed to sell the branches which the European Union had demanded should be disposed of in return for £65bn of state aid.

The specific capital shortfalls of Lloyds and RBS were not disclosed but are thought to make up a large part of the £25bn hole identified by the the Bank of England's financial policy committee in March.

The IMF said the new stress tests by the PRA planned for 2014, following this year's exercise for the financial policy committee, should provide more detail about the methodology, results and bank-by-bank capital plans.

Lloyds, which analysts estimate has a £3bn shortfall, said it could plug its gap by generating profits and continuing to sell non-core assets, such as problem loans – ensuring taxpayers and other investors would not need to buy new shares or other types of financial instruments.

To underline the point, Lloyds raised £500m just after the stock market closed by selling another tranche of its stake in wealth manager St James's Place. Since March, Lloyds' stake in the firm has fallen from 57% to 21%.

RBS said it could fill its capital shortfall by selling off part of its US business, Citizens, and scaling back its investment bank.

The PRA said that further announcements would come from other banks once discussions over capital had been concluded. The City is most concerned about the outcome of discussion with Co-operative Bank.